if eps < -1.9234536f-05 or 4.8450306f-11 < eps

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   13.9
2. Using strategy rm
   13.9
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   0.6
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   0.6
5. Using strategy rm
   0.6
6. Applied flip-+ to get
   \[\color{red}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)} \leadsto \color{blue}{\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}}\]
   0.6
7. Applied simplify to get
   \[\frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{red}{\sin x \cdot \cos \varepsilon - \left(\cos x \cdot \sin \varepsilon - \sin x\right)}} \leadsto \frac{{\left(\sin x \cdot \cos \varepsilon\right)}^2 - {\left(\cos x \cdot \sin \varepsilon - \sin x\right)}^2}{\color{blue}{(\left(\cos \varepsilon\right) * \left(\sin x\right) + \left(\sin x\right))_* - \sin \varepsilon \cdot \cos x}}\]
   0.7

if -1.9234536f-05 < eps < 4.8450306f-11

1. Started with
   \[\sin \left(x + \varepsilon\right) - \sin x\]
   20.6
2. Using strategy rm
   20.6
3. Applied sin-sum to get
   \[\color{red}{\sin \left(x + \varepsilon\right)} - \sin x \leadsto \color{blue}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right)} - \sin x\]
   15.0
4. Applied associate--l+ to get
   \[\color{red}{\left(\sin x \cdot \cos \varepsilon + \cos x \cdot \sin \varepsilon\right) - \sin x} \leadsto \color{blue}{\sin x \cdot \cos \varepsilon + \left(\cos x \cdot \sin \varepsilon - \sin x\right)}\]
   15.0
5. Using strategy rm
   15.0
6. Applied log1p-expm1-u to get
   \[\sin x \cdot \cos \varepsilon + \left(\color{red}{\cos x \cdot \sin \varepsilon} - \sin x\right) \leadsto \sin x \cdot \cos \varepsilon + \left(\color{blue}{\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*)} - \sin x\right)\]
   15.0
7. Applied taylor to get
   \[\sin x \cdot \cos \varepsilon + \left(\log_* (1 + (e^{\cos x \cdot \sin \varepsilon} - 1)^*) - \sin x\right) \leadsto \log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)\]
   0.1
8. Taylor expanded around 0 to get
   \[\color{red}{\log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)} \leadsto \color{blue}{\log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*)}\]
   0.1
9. Applied simplify to get
   \[\log_* (1 + (e^{\sin \varepsilon \cdot \cos x} - 1)^*) \leadsto \cos x \cdot \sin \varepsilon\]
   0.1

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10. Applied final simplification